Construction of a field with roots of unity in MAGMA

364 Views Asked by At

I would like to construct a field (NumberField) that contains a (primitive) $n$-th root of unity and $i = \sqrt{-1}$ using a computer algebra system MAGMA, i.e. $\mathbb{Q}(\zeta_n, i)$.

I tried constructing it as follows (for $n=7$):

n := 7;
Qx<x> := PolynomialRing(Rationals());
f := CyclotomicPolynomial(n);
g := x^2+1;
F<a> := SplittingField([f,g])
DefiningPolynomial(F);

Here the element a gives me the $n$-th root of unity. But how do I get b such that $b^2=-1$?

Thanks in advance! -- Mike

P.S. If someone wants to test MAGMA, you can try online here.